ANALYSIS AND DESIGN OF SWITCHING AND FUZZY
SYSTEMS
a dissertation
submitted to the department of electrical and electronics
engineering
and the institute of engineering and sciences
of bilkent university
in partial fulfillment of the requirements
for the degree of
doctor of philosophy
By
Murat Akg¨
ul
September 2002
I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Doctor of Philosophy.
¨
Omer Morg¨ul, Ph. D. (Supervisor)
I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Doctor of Philosophy.
B¨ulent ¨Ozg¨uler, Ph. D.
I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Doctor of Philosophy.
I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Doctor of Philosophy.
H. Altay G¨uvenir, Ph. D.
I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Doctor of Philosophy.
Aydan Erkmen, Ph. D.
Approved for the Institute of Engineering and Sciences:
Prof. Dr. Mehmet Baray
ABSTRACT
ANALYSIS AND DESIGN OF SWITCHING AND FUZZY
SYSTEMS
Murat Akg¨
ul
Ph.D. in Department of Electrical and Electronics Engineering
Supervisor: Prof. Dr. ¨
Omer Morg¨
ul
September 2002
In this thesis we consider the controller design problems for switching and fuzzy systems. In switching systems, the system dynamics and/or control input take different forms in different parts of the underlying state space. In fuzzy systems, the system dynamics and/or control input consist of certain logical expressions. From this point of view, it is reasonable to expect certain similarities between these systems. We show that under certain conditions, a switching system may be converted into an equivalent fuzzy system. While the changes in the system variables in a switching system may be abrupt, such changes are typically smooth in a fuzzy system. Therefore obtaining such an equivalent fuzzy system may inherit the stability properties of the original switching system while smoothing the system dynamics. Motivated from this idea we propose various switching strategies for certain classes of nonlinear systems and provide some stability results. Due to the difficulties in designing such switching rules for nonlinear systems, most of the results are developed for certain specific type of systems. Due to the logical structure, obtaining rigorous stability results are very difficult for fuzzy systems. We propose a fuzzy controller design method and prove a stability result under certain conditions. The proposed method may also be applied to function approximation. We also consider a different stabilization method, namely phase portrait matching, in which the main aim is to choose the control input appropriately so that the dynamics of the closed-loop system is close to a given desired dynamics. If this is achieved, then the
phase portrait of the closed-loop system will also be close to a desired phase portrait. We propose various schemes to achieve this task.
Keywords : Fuzzy Systems, Switching Systems, Stabilization, Controller Design, Driftless Systems, Phase Portrait Matching, Lyapunov Functions, Periodic Switching.
¨
OZET
ANAHTARLAMALI VE BULANIK S˙ISTEMLER˙IN ANAL˙IZ
VE TASARIMI
Murat Akg¨
ul
Elektrik ve Elektronik M¨
uhendisli˘gi Doktora
Tez Y¨oneticisi: Prof. Dr. ¨
Omer Morg¨
ul
A˘gustos 2002
Bu tezde, anahtarlamalı ve bulanık sistemler i¸cin denetleyici tasarım problemlerini ele alıyoruz. Anahtarlamalı sistemlerde, sistem dinami˜gi ve/veya denetleyici girdisi, durum uzayının de˜gi¸sik b¨olgelerinde de˜gi¸sik bi¸cimler almaktadır. Bulanık sistemlerde, sistem dinami˜gi ve/veya denetleyici girdisi belirli bazı mantıksal ifadelerden olu¸smaktadır. Bu a¸cıdan bakıldı˜gında, sistemler arasında belli benzerlikler olmasını beklemek makul g¨or¨ulmektedir. Belli ko¸sullar altında, anahtarlamalı bir sistemin e¸sde˜ger bir bulanık sisteme d¨on¨u¸st¨ur¨ulebilece˜gini g¨osteriyoruz. Sistem parametrelerinin de˜gi¸smesi anahtar-lamalı sistemlerde ani olur iken, benzer de˜gi¸simler bulanık bir sistem i¸cin genellikle d¨uzg¨un/kesiksiz olmaktadır. Dolayısıyla, bu ¸sekilde elde edilen bir bulanık sistem, orjinal anahtarlamalı sistemin kararlılık ¨ozelliklerini g¨osterirken aynı zamanda sistem dinami˜gini d¨uzg¨unle¸stirebilir. Bu d¨u¸s¨unceden hareketle, do˜grusal olmayan belli bir sınıf sistem i¸cin, de˜gi¸sik anahtarlama y¨ontemleri ¨oneriyor ve bazı kararlılık sonu¸cları g¨osteriyoruz. Do˜grusal olmayan sistemler i¸cin benzer anahtarlama kurallarının tasarımındaki zorluk dolayısıyla, geli¸stirilen sonu¸cların ¸co˜gu belirli tip sistemler i¸cin olmaktadır. Bulanık sistemler i¸cin genel bir kararlılık sonucu elde etmek, mantıksal yapısı nedeniyle olduk¸ca zordur. Bulanık denetleyiciler i¸cin bir tasarım y¨ontemi ¨oneriyor ve belirli ko¸sullar altında kararlılı˜gını g¨osteriyoruz. ¨Onerilen y¨ontem fonksiyon yakınsama i¸cin de uygulanabilir. Ayrıca faz portresi e¸sleme ismiyle, amacı kapalı d¨ong¨u sistem dinami˜ginin verilen bir dinami˜ge m¨umk¨un oldu˜gunca yakınsamasını sa˜glayacak uygun kontrol girdisinin
hesaplanması olan, farklı bir kararla¸stırma y¨ontemini ele aldık. E˜ger bu sa˜glanabilirse, kapalı d¨ong¨u sistemin faz portresi istenilen bir faz portresine yakınsanabilir. Bunun i¸cin de˜gi¸sik tasarımlar ¨oneriyoruz.
Anahtar Kelimler : Bulanık Sistemler, Anahtarlamalı Sistemler, Kararlılık, Denetleyici Tasarımı, S¨ur¨uklenmesiz Sistemler, Faz Portresi E¸sleme, Lyapunov Fonksiy-onları, Periyodik Anahtarlama.
ACKNOWLEDGEMENTS
First, I want to express my deep gratitude to my parents, grandmother and brothers, who have always been there behind me with their love, support, encouragement and prayers during the hard times of my PhD adventure. I dedicated this thesis to my dear parents.
I would like to thank my supervisor Prof. Dr. Omer Morg¨¨ ul for his invaluable guidance, and patience. He has always been a kind and motivating advisor. I want to mention about his guidance especially on how to write for possible readers, rather than myself. I should acknowledge his encouragements and supports when I get stuck and unable to see the light.
I want to express my special thanks to Prof. B¨ulent ¨Ozg¨uler, Prof. Enis C¸ etin, Prof. H. Altay G¨uvenir and Prof Aydan M. Erkmen for reading and commenting on the thesis. I would also express my sincere thanks to Prof. Aydan M. Erkmen for the short but guiding discussions.
During these years, Hakan K¨oro˜glu and Ercan Solak have been close friends. They were the two who contributed to me most in our thoughtful technical and nontechnical discussions. Nejat Kamacı, Mustafa Sungur, Engin Avcı, H. Bayram Erdem, ¨Omer Saliho˜glu have been amiable housemates. I would like to thank to M.B.Y. for her constant love, patience, and support during these years. There has been many other friends, a partial list of them being Ay¸seg¨ul S¸ahin∗, Elif ¨O˜g¨ut, K. G¨u¸cl¨u K¨opr¨ul¨u, Karim Saadaoui,
L¨utfiye Durak, Mustafa Akba¸s, Necmi Bıyıklı, Tolga Kartalo˜glu, and Z. G¨urkan Figen with whom I enjoyed life in Bilkent. I want to take this opportunity to express my thanks to all of them.
Finally, I would like to thank my department and the whole Bilkent community for supplying us such a distinguished university environment.
Dedicated to
Naciye and H¨
useyin
Akg¨
ul
Contents
1 Introduction 1
1.1 Switching Systems . . . 3
1.2 Fuzzy Systems . . . 4
1.3 Organization of the Thesis . . . 5
2 Switching Systems 8 2.1 Classical Set Theory . . . 10
2.1.1 Operations on Sets . . . 11
2.1.2 Relations . . . 12
2.1.3 Implications . . . 13
2.2 Models for Switching Systems . . . 14
2.3 Stability Analysis . . . 17
2.3.1 Commuting Stable Matrices . . . 18
2.3.2 Multiple Lyapunov Functions . . . 19
2.3.3 Other Stability Results . . . 20
3 Fuzzy Systems 22 3.1 Fuzzy Set Theory . . . 25
3.1.1 Operations on Fuzzy Sets . . . 28
3.1.2 Fuzzy Relations . . . 30
3.1.3 Fuzzy Logic and Approximate Reasoning . . . 33
3.2 Models for Fuzzy Systems . . . 34
3.2.1 Mamdani Type Fuzzy Systems . . . 36
3.2.2 Takagi-Sugeno Type Fuzzy Systems . . . 37
3.3 Stability Analysis . . . 38
3.3.1 Mamdani Type Fuzzy System . . . 39
3.3.2 Takagi-Sugeno Type Fuzzy System . . . 40
3.3.3 Other Stability Results . . . 41
4 Comparison of Switching and Fuzzy Systems 43 4.1 Comparison of the Models . . . 44
4.1.1 Simulation . . . 47
4.2 Comparison of the Stability Results . . . 50
5 Phase Portrait Matching 52 5.1 Minimization of the Phase Difference . . . 53
5.1.1 Problems . . . 54
5.1.2 Solution when n = m+1 . . . 56
5.1.3 Projection Operators . . . 57
5.1.4 Other Solutions . . . 59
5.1.5 Saturation of the Control Signal . . . 63
6 Fuzzy Controller and System Design 69
6.1 General Design Aspects . . . 70
6.1.1 Fuzzy Controller Design Parameters . . . 71
6.1.2 Fuzzy Controller Design Guidelines . . . 72
6.2 Fuzzy Controller Design for Systems with Completeness Property . . . . 74
6.2.1 Problem Statement and Definitions . . . 75
6.2.2 Stability Analysis . . . 75
6.2.3 Design Procedure . . . 78
6.3 Fuzzy Controller Design Using Data Mining . . . 81
6.3.1 Calculation of eK(x) . . . 82
6.3.2 Calculation of Ki . . . 83
6.3.3 Calculation of Vertices, bKi . . . 85
6.3.4 Calculation of µSi(x) . . . 88
7 Switching Controller and System Design 89 7.1 Switching Strategies . . . 90
7.1.1 Dwell Time Switching . . . 90
7.1.2 Supervisory Switching . . . 92
7.1.3 Logic Based Switching . . . 92
7.2 A Switching Synchronization Scheme for a Class of Chaotic Systems . . . 93
7.2.1 A Brief Review of Chaos . . . 95
7.2.2 Problem Statement . . . 96
7.3 Periodic Switching . . . 103
7.4 A Switching Strategy Emulating Common Lyapunov Function . . . 105
7.4.1 Event Driven Switching . . . 107
7.5 A Switching Strategy to Stabilize some of the Driftless Systems . . . 109
8 Applications 112 8.1 System Modelling . . . 113
8.1.1 Using Switching System . . . 113
8.1.2 Using Fuzzy Controller Obtained by Data Mining . . . 117
8.2 Observer Based Chaotic Message Transmission . . . 119
8.2.1 Observer Based Chaotic Masking Scheme . . . 119
8.2.2 An Observer Based Chaotic Modulation Scheme . . . 122
8.3 Stabilization . . . 126
8.3.1 Using PPM . . . 126
8.3.2 Using Switching . . . 129
8.3.3 Using Fuzzy Controller for Systems with Completeness Property . 139 9 Conclusions 142 9.1 Results and Contributions . . . 142
9.1.1 Modelling . . . 142
9.1.2 Phase Portrait Matching . . . 143
9.1.3 Stabilization . . . 144
9.2 Research Directions . . . 145 9.2.1 Obtaining an Equivalent Switching System for a Given Fuzzy System145
9.2.2 Design of the Desired Phase Portrait . . . 145 9.2.3 Fuzzy Controller Design . . . 146 9.2.4 Switching Controller Design . . . 147
List of Figures
3.1 The general structure of a fuzzy systems. . . 25 3.2 An example of a relationR between u and v. . . 30 3.3 (a)Projection of the R on V, (b) Cylindrical extension of S into U. . . . 32 3.4 The fuzzy setB, obtained by the projection of the composition of A with
R. . . 33 3.5 The general structure of a fuzzy systems. . . 35
4.1 The functions defining the boundaries of the regions Ri (a), and the
functions ˜Sj(x), i = 1, 2, 3(b)(c)(d) . . . 48
4.2 Control signals for switching and the corresponding fuzzy system . . . 50 4.3 States of the switching and the corresponding fuzzy system. . . 50
5.1 The characteristic of an affine system at some points x where (a)β > 0 (b)β < 0. . . 54 5.2 The situation with small phase difference between f (x) and g(x) when
(a)β > 0 (b)β < 0. . . 55 5.3 The range of ˙x vectors from f (x) + g(x)Umin to f (x) + g(x)Umin when (a)
β > 0 (b) β < 0. . . 56 5.4 An affine system with n = 3 and m = 2. . . 59 5.5 An affine system with constraints on u(x). . . 64
5.6 Phase matched saturation of u(x) with planar boundaries. . . 64
5.7 Phase matched saturation of u(x) with circular boundaries. . . 66
6.1 Control system with negative unit feedback. . . 70
6.2 The control gain ˜K(x) with minimum norm. . . 83
6.3 The ball BR inside the convex hull of Ki(x) for n = 2. . . 84
6.4 The ball BR inside the convex hull of K(x) for n = 3. . . 87
8.1 Simulation result for Chua’s circuit (a) z1 vs z2, (b) z1 vs. z4 = ˆz1, (c) z2 vs. z5 = ˆz2, (d) kek vs time. . . 114
8.2 Simulation result for Brockett system (a) z1 vs z2, (b) kek vs time, (c) e1 = z1− ˆz1 vs time for a = 0.2, (d) e1 = z1− ˆz1 vs time for a = 0.02. . . 116
8.3 The membership function of the fuzzy system approximating f (z, t). . . . 118
8.4 (a) The output of the fuzzy system, and (b) the error made in the modelling.118 8.5 (a) The transmitted signal y = z1+ m (b) the graph of z1 vs z2 (c) the message m(t) (d) the recovered message mr(t) . . . 122
8.6 (a) The transmitted signal z1(t) (solid) and the message m(t), (b) z1 vs z2 125 8.7 (a) The stabilization of Lorenz chaotic system using norm minimization and (b) the corresponding control signals. . . 128
8.8 (a) The stabilization of Lorenz chaotic system using phase difference minimization and (b) the corresponding control signals. . . 128
8.9 States of the chained system stabilized using periodic switching. . . 132
8.10 Stabilization of nonholonomic integrator using sequentially stabilizing the states. (a) States vs. time (b) kxk vs. time. . . 134
8.11 Stabilization of nonholonomic integrator starting from the state with the highest absolute value. (a) States vs. time (b) kxk vs. time. . . 134
8.12 Stabilization of Lorenz Chaotic system, starting from the state with the highest absolute value(λ1 = 100). (a) States vs. time (b)kxk vs. time. . 135
8.13 Stabilization of Lorenz Chaotic system, starting from the state with the highest absolute value (λ1 = 4). (a) States vs. time (b) kxk vs. time. . . 136
8.14 Stabilization of unicycle starting from the state with the highest absolute value: (a) States vs. time (b) The trajectory of the unicycle. . . 137 8.15 Stabilization of chained systems using the properties of the states being
distinctly stabilizable. (a) States vs. time (b) kxk vs. time. . . 138 8.16 Membership functions of the Fuzzy Logic Controller. . . 139 8.17 (a) Sum of the membership functions and the domain of attraction, (b)
The phase portrait of the closed-loop system and the simulations. . . 140 8.18 The phase portrait of the closed-loop system and the simulations. . . 141
List of Tables
2.1 Truth table for p implies q . . . 14
List of Acronyms
FBF : Fuzzy Basis Function.
FLC : Fuzzy Logic Controller.
FLS : Fuzzy Logic System.
GMP : Generalized Modus Ponens.
GMT : Generalized Modus Tollens.
LHP : Left half plane.
LTI : Linear Time Invariant.
MF : Membership Function.
ODE : Ordinary Differential Equation.
PD : Proportional Derivative.
PID : Proportional Integral Derivative (Controller).
RPM : Revolution Per Minute.
SS : Switching System.
Chapter 1
Introduction
There is a continuous search for the models of the phenomena occurring around us. The understanding of the environment and its dynamics is one of the challenges to the human life to carry on. Therefore we try to associate a model with each phenomenon we see around us. The closer the model to the real world the better results we obtain in interacting with it. The rainfall pattern throughout a year in a country is an example of such a phenomenon. This local model is vital for the farmers in that country. Another example would be cattle dealing. The knowledge of the animal requirements like, feeding, breeding, and health care are components of the model that affects the output of the process, cattle dealing. Generally a human is in the position to control certain aspects of the phenomenon under consideration. The knowledge, gathered as the experience of years, is the model which can be expressed in general by subjective expressions.
Increase in the population resulted in the problem of increasing the production efficiently to meet the resulting demand. One of the solution is to use machines in the production lines to increase the efficiency. This tendency brings the need for engineers that can build machines which automatically perform certain tasks. This speeds up the advances in the control theory, concerning with the development of strategies which force a given system to exhibit desirable behavior. The systems considered might be physical, chemical, biological or even social in nature. The design attempts start with the modelling of the system, which is usually referred to as the plant [1]. In general, the model might be expressed with nonlinear differential equations.
Consider the problem of designing an automatic camera that can adjust its parameters depending on the distance of the target, the illumination of the environment, etc. One good starting point is to find some professional photographers and ask how s/he would decide on these adjustments. Or, consider to design an auto-pilot for an automobile. Again the knowledge of the drivers of that specific car model will be useful to start the design. Since these experts generally will not be able to express their experience in a differential equation form, all the information the interviewer will obtain is a collection of sentences. Generally these sentences will be conditionals like IF “certain conditions exist” THEN “perform certain actions”. However, in general the conditional parts of the statements or the consequent parts might not be expressed so precisely, e.g. IF “the road is slippery due to the rain” THEN “don’t exceed 30km/h”. Such conditions are vague and require further effort to map them into the measurable domains.
If the given conditions can be measured precisely and the control action is also expressed precisely then one might come up with a controller structure that switches in between the specified control actions depending on the measured conditions. This type of controllers are known as switching controllers in the literature. The tools used in the design of such systems are the control theory developed so far plus the classical set theory and logic for the interpretation of the given rules. On the other hand if expressions are vague in nature and/or subjective then these expressions first must be mapped onto the measurable domains so that the interfaces between the sensors and actuators can be established. In the literature, the fuzzy set theory and the fuzzy logic provide the necessary tools to interpret these vague expressions and the conditional statements. Fuzzy control theory is an approach that utilizes fuzzy logic and the control theory tools to build controllers based on such vague specifications.
Many nonlinear systems can be approximated as a sum of piecewise linear systems. Hence a switching system that switches between these linear models will be a good approximation for the nonlinear model. The controller design for Linear Time Invariant (LTI) systems is one of the subject that has almost reached its mature state in the literature. The controllers designed for each of the linear systems, which constitute the nonlinear model, will be the switching controller to improve the performance of the system under consideration. Following the same reasoning fuzzy systems can also be
used. Their use, may improve the performance of a designed switching controller such as smoothing some of the system variables.
In this thesis we consider two logic based system design approaches, namely the switching systems and the fuzzy systems. We investigate the modelling of both approaches and seek some methods to convert one model to the other so that the results obtained for one model can be applied to the other model. Fuzzy systems are known to be universal approximators. Hence any model can be approximated by a fuzzy system up to a given accuracy. One of the problems in control theory is the stabilization of a given plant. So we can conclude that if a system is stabilizable by a controller then there should be a fuzzy system which can approximate it for a given accuracy. Another problem we investigate in this thesis is to find a method to approximate a given input/output data by a fuzzy system. The determination of an input/output data is in fact another problem. We try to solve this problem by using phase portrait approach. The other topics we study studied in this thesis can be summarized under the topic of modelling and stabilization problems of switching and fuzzy systems.
In the following section we will give a brief literature survey on the switching and fuzzy systems and give some motivation on the subject. Then we will conclude this chapter with the organization of the thesis.
1.1
Switching Systems
The idea of switching is well known in control theory. In past decades, switching has been used in adaptive control to assure stability. In [2] switching between multiple adaptive models was used to improve the transient response of adaptive control systems in a stable fashion [3]. There are several other reasons for switching control to become popular. Any nonlinear system can be modelled with a linear system in a certain region by linearization. A switching system whose dynamics switches in between these linearized models will approximate the dynamics of the nonlinear system. The controllers which is designed for the linearized models will constitute the switching controller for the nonlinear system.
Motivated by the current interest in switching systems several authors have studied optimal control [4] and stability questions for switching systems [5]- [6]. More details about these studies are provided in Section 2.
In most of the works related with switching systems, the concept of a common Lyapunov unction is utilized to obtain some stability results. The necessary and sufficient conditions which guarantee the existence of a common Lyapunov function are not known, the problem may even be undecidable (see [7]). Hence many of the current work on this area focus on special cases, and such studies lead to special switching strategies to stabilize such systems.
In this thesis our main concern related with switching systems is the determination of a stabilizing control signal for a class of nonlinear systems. The design methodology and modelling aspects constitute a base for the models of fuzzy systems.
1.2
Fuzzy Systems
A fuzzy system is a real time expert system implementing a part of a human operator’s or process engineer’s expertise which does not lend itself to being easily expressed in a difference/differential equation but rather in situation/action rules like the IF-THEN statements given previously.
It is generally agreed that an important point in the evolution of the modern concept of uncertainty was the publication of a seminal paper by Lotfi A. Zadeh [8]. In this paper, Zadeh introduced a theory whose objects-fuzzy sets-are sets with boundaries that are not precise. The membership in a fuzzy set is not a matter of affirmation or denial, but rather a matter of degree.
The first successful industrial application of the Fuzzy Logic Controller (FLC) was a cement kiln control system developed by the Danish cement plant manufacturer F. L. Smidth in 1979.
A representation theorem, mainly due to Kosko [9], states that any continuous nonlinear function can be approximated as closely as needed with a finite set of fuzzy
variables, values, and rules. Later X.-J. Zeng and M. G. Singh published their work on the approximation problem of SISO and MIMO fuzzy systems (see [10], [11]). These results describe the representational power of fuzzy control in principle, but it does not answer the questions of how many rules are needed and how they can be found.
FLCs can be used in various ways. One of them is to use the FLC directly as the controller for the system under consideration. The other choice is to use a standard controller like PID or lead-lag and tune the parameters of this controller with a Fuzzy Logic System (FLS). More details on these studies can be found in Section 3. Our main concern on this subject is the design methodologies that results in a fuzzy controller providing closed-loop stabilization.
1.3
Organization of the Thesis
In this thesis we consider the modelling and control problems of switching and fuzzy systems. The chapters and their contents can be summarized as follows.
Chapter 2: Switching Systems
In this chapter we have three main topics: Classical set theory, models for switching systems, and stability analysis. In classical set theory section, we give the necessary tools like, operations on sets, relations, and interpretation of conditional statements, to model a switching system. In models for switching systems section, we consider the various forms of models suggested in the literature. In stability analysis section, we give the basic stability results in the literature some of which will be utilized in Chapter 7.
Chapter 3: Fuzzy Systems
This chapter is composed of three main topics: Fuzzy set theory, models for fuzzy systems, and stability analysis. Fuzzy set theory section provides the necessary tools like operations on fuzzy sets, fuzzy relations, and interpretation of conditional statements with fuzzy propositions, to model a fuzzy system. The models for fuzzy systems section
provides various forms of models suggested in the literature. In stability analysis section, we give the basic stability results available in the literature.
Chapter 4: Comparison of Switching and Fuzzy Systems
In this chapter we investigate the resemblance of switching and fuzzy system regarding the model structure and the stability results available in the literature. In discussing the modelling issues we propose a method to obtain an fuzzy system associated with a given switching system.
Chapter 5: Phase Portrait Matching
In this chapter we propose a method to calculate the control signal of a given system so that the closed loop dynamics is close to a desired dynamics. We devise two approaches to calculate the control signal. The first method aims to minimize the phase difference between the desired phase portrait and the closed-loop system. The second method aims to minimize the norm of the difference of the desired dynamics and the dynamics of the closed-loop system. The control signal calculated by these schemes might be out of the physical bounds of the given system. We propose two saturation schemes which might be used when the calculated control signal magnitude is out of some prescribed limits.
Chapter 6: Fuzzy Controller and System Design
In this chapter we have three main sections. In the first section we give the general design aspects for fuzzy controllers. In the remaining sections of this chapter we propose two methods for the design of fuzzy controllers. We show that the method we propose in the second section yields a stable closed-loop system under certain conditions. We also outline a design procedure related with the proposed method. In the third section we propose a method to approximate the calculated control signal by using fuzzy rules. If the input/output data is known for a stabilizing controller then this method can also be used to construct a fuzzy controller to be used in the stabilization problem.
Chapter 7: Switching Controller and System Design
This chapter is composed of 5 sections. In the first section we provide the tools available in the literature to design a switching system or controller. In the second section we consider a chaotic system which has piecewise linear dynamics. We model this system as a switching system, propose a synchronization scheme and give various stability results related with the proposed scheme. In the third section we propose a periodic switching strategy to stabilize a class of nonlinear systems. In the fourth section we propose another method for stabilization of a class of nonlinear system using a switching strategy so that the closed loop system becomes stable. In the fifth section we propose a switching strategy to stabilize a certain class of driftless systems.
Chapter 8: Applications
In this chapter we present some of the applications and simulation results related to the methods we proposed in the thesis. In the first section we present some simulation results related to the chaotic synchronization scheme given in Section 7.2. We also present some simulation results related with the fuzzy approximation technique we proposed in Section 6.3. In the second section, we propose two chaotic message transmission schemes for a class of switching chaotic systems, prove some convergence results. We also present some simulation results related to the proposed chaotic masking scheme. In the third section, we present some simulation results related to the methods we proposed for the stabilization of switching and fuzzy systems. We also present some stability results related with the switching schemes we proposed for a class of systems.
Chapter 2
Switching Systems
Many complicated systems, such as airplanes manufacturing and transportation systems etc, in general have many operating points, each of which should be handled carefully. We also see that today’s products incorporate logical decision-making into even the simplest control loops (e.g., embedded systems) [12]. These systems are generally modelled almost at every operating point and a suitable control action is applied accordingly. Hence in such systems, both the system model used in designing controllers, and the control action changes, or switches, depending on operating points. Such systems are natural examples of switching systems. Formal definition of switching systems will be given later, see Section 2.2, but at this point we may state that by switching systems we mean a system for which the system dynamics and/or control action changes (or switches) according to a rule. In [13] there has been made distinction between the switching systems and the hybrid systems. A hybrid system is defined as being one that the switching rule considers the history of previous modes while the switching systems just checks some conditions at that time and discards the past modes of the system. There are studies where this distinction is omitted as well, see [7]. Although in our study the rule of the switching system is a function of only the current states of the switching system we accept to have a system with a switching rule which might be a function of the switching history, see e.g. (2.1).
The idea of switching is well known in control theory. In past decades, switching has been used in adaptive control to assure stability. In [2] switching between multiple
adaptive models was used to improve the transient response of adaptive control systems in a stable fashion [3]. There are several other reasons for switching control to become popular. Any nonlinear system can be modelled with a linear system in a certain region by linearization. A switching system whose dynamics switches in between these linearized models will approximate the dynamics of the nonlinear system. The controllers which is designed for the linearized models will constitute the switching controller for the nonlinear system. A desired performance for the system might require too many number of controllers, which in return requires a high computational power. In fact the improvements which puts a high computational power to a small chip and the recent developments in sensor and actuator technology are the main fuels of the tendency to switching systems. By the sensors it is possible to measure the mode of the system and by the computers it is possible to decide in real-time how to feed the actuators to force the system to have an acceptable performance.
Motivated by the current interest in switching systems several authors have studied optimal control [4] and stability questions for switching systems [5], [14], [15], [16], and [6]. In [15], an exactly known linear systems and a given set of control laws are considered, and a method to determine whether the closed-loop system is stable under all possible switching sequences, is presented. In [5] the idea of multiple Lyapunov functions as a tool for studying the stability of switching systems is introduced, and some existence results for a set of controllers to be stabilizing in the sense of Lyapunov are presented. The stabilization of a linear plant by switching two linear control laws is considered in [14] and a method for determining a stabilizing switching sequence is described. In [16] a sufficient condition for robust output feedback stabilization with synchronous controller switching is presented. The necessary and sufficient conditions to test for quadratic stability and stabilizability and for stabilizability with a quadratic storage function for switching controller systems are presented in [6].
Most of these works utilize the concept of a common Lyapunov function to study stability of switching systems. The necessary and sufficient conditions which guarantee the existence of a common Lyapunov function are not known, the problem may even be undecidable (see [7]). Hence many of the current work on this area focus on special cases, and such studies lead to special switching strategies to stabilize such systems.
This chapter is organized as follows. In Section 2.1 we will review shortly the classical set theory. This would be constructive for the following sections and chapters. In Section 2.2 the modelling issues of switching systems will be given. In Section 2.3 the stability results in the literature will be reviewed.
2.1
Classical Set Theory
A classical set is a collection of objects of any kind. The concept of a set has become one of the most fundamental notions of mathematics. So-called set theory was founded by the German mathematician George Cantor (1845-1918). In set theory the notions ’set’ and ’element’ are primitive. They are not defined in terms of other concepts. Letting A be a set, “x∈ A” means that x is an element of the set A and “x /∈ A” means that x does not belong to the set A. The way in which these elements are specified is immaterial: for example, there is no difference between the set consisting of the elements A ={2, 3, 5, 7} and B = {the set of all prime numbers less then 11} [17]. Given a certain property, (e.g. having four legs), an interesting question is the following: Does there exist a set whose elements are exactly those having the given property (e.g. having four legs)? We are inclined to answer affirmatively to this question. However accepting an affirmative answer to this question might lead to different versions of Russell’s Paradox. For this reason mathematicians have confined each of their discussions to some universal set or universe of discourse [18].
For any element x and a set A in the universe of discourse X it can unambiguously be determined whether x ∈ A or x /∈ A. A classical set may be finite, countable or uncountable. It can be described either by listing up the individual elements of the set or by stating a property for the membership. The set C ={red, orange, yellow, green, blue} is an example of a finite set that is described by its elements. The set T ={x ∈ Z|x ≥ 0}, is an example of countable set that is described by a property. The real interval [0, 1] is an example of an uncountable set. Two sets are very important, namely, the universe X, containing all elements of the universe of discourse, end the empty set ∅, containing no elements at all.
If P (x) is a predicate stating that x has a property P , then a set can also be denoted by{x|P (x)}. This leads us to a third way of defining a set A, which utilizes the concept of characteristic function µA(·) given below.
Definition 1 Characteristic Function : Let A be defined in a universe of discourse X.
µA : X → {0, 1}
is a characteristic function of the set A iff the following holds for all x:
µA(x) = 1, x∈A 0, x /∈A.
Note that the output of the predicate P (x) is Boolean, that is either TRUE or FALSE, whereas the output of a characteristic function µA is {0, 1}. These two functions can
be related with the following expression: P (x) ⇔ (µP(x) = 1) [17]. Also note that this
definition of a classical set has a natural extension to the definition of a fuzzy set, see Section 3.1.
In the subsequent sections we will give some of the definitions for the set operations, define the relations and briefly define implications. These definitions will be modified in Section 3.1 using characteristic functions.
2.1.1
Operations on Sets
Classical set theory uses several operations like complement, intersection, union etc. The following definitions give the description of the operators which can be defined over set(s).
Definition 2 Set Operations :
Let A and B be two classical sets in a universe of discourse X. Then various set operations can be defined as follows:
Complement of A, A0 = {x|x /∈ A}
Intersection of A and B, ATB = {x|x ∈ A and x ∈ B} Union of A and B, ASB = {x|x ∈ A or x ∈ B}
Difference of A from B, A− B = {x|x ∈ A or x /∈ B} Symetric difference of A and B, A + B = (A− B)S(B− A) Power set of A, P (A) = {U|U ⊆ A}
Cartesian product of A and B, A× B = {(x, y)|x ∈ A and x ∈ B} Power n of A, An = A× · · · × A
| {z }
ntimes
There are some useful properties of these set operators which have been omitted. Interested readers can check any elementary set theory books like [17], [18].
2.1.2
Relations
A relation R can be considered as a set of tuples, where the tuples are ordered pairs. A binary (dyadic) tuple is denoted by (x, y), an example of ternary tuple is (x, y, z), and an example of n-ary tuple is (x1, . . . , xn). Let X be the domain
of people, i.e. X = { Figen, Murat, Filiz, Timur, Sel¸cuk} and let Y be the domain of height, e.g. Y = {1.65, 1.74, 1.78, 1.80, 1.82}. Then a relation which can be characterized as ”has a height of” defined on X × Y is given as R = {(Figen, 1.65), (Murat, 1.78), (Filiz, 1.74), (Timur, 1.80), (Sel¸cuk, 1.82)}. Consider the domain of natural numbers, N. Then “ ≤ ” is a relation on N × N and defined as {(m, n)|m ≤ n}. Clearly (4, 1) is not in this relation. As should be noted this relation is a subset of N× N. If we compare the definition of a set using predicates and those for a relation given above, we can say that a relation is a set in a product space. Hence just like classical sets, classical relations can be described by characteristic functions.
Definition 3 Relation :
Let a relation R be defined in a universe of discourse X = X1× · · · × Xn.
µR : X1× · · · × Xn→ {0, 1}
is a characteristic function of R iff the following holds for all x = (x1, . . . , xn)∈ X.
µR(x) = 1, (x1, . . . , xn)∈R 0, (x1, . . . , xn) /∈R
Relations can have several properties like reflexive, symmetric, or transitive. We exclude their definitions and leave it to the reader to check the references [17], [18] for more details.
2.1.3
Implications
Making statements is a basic human activity; it is part of that complex activities that collectively we call language. A fundamental property of statements is that they may be true or false; which is called the truth value of the given statement. Typically, a statement is understood as saying something about the world, and its truth value is assessed on that basis. If the world is in fact the way a statement says it is, then we call that statement TRUE; if not, we call it FALSE. There can be statements where the truth value can not be judged without supplementary information. Let us call these kind of statements as open propositions for the motivations of the subsequent discussions.
Definition 4 Variable, Replacement Set :
A variable is a symbol which represents an unspecified or arbitrary element of a specified set. Such a set is also called the replacement set.
Definition 5 Open Proposition :
An open proposition is a declarative sentence which
(a) contains a finite n number of variables, where n≥ 1, (b) is neither true nor false, but
(c) becomes TRUE or FALSE when the variables are replaced by elements from their replacement set.
Let x represent an element of the set C, where C is the set of names of capital cities in the world and y represents an element of the set S, where S is the set of names of the states in the world. Then the following is an example for an open proposition;
p q p ⇒ q 0 0 1 0 1 1 1 0 0 1 1 1
Table 2.1: Truth table for p implies q
Any proposition can also be related to each other. Sometimes two proposition are so related that the first cannot be TRUE without the second being TRUE as well. For example “A linear system ˙x = Ax is stable if and only if all eigenvalues of A has negative real parts.”. There are also conditional propositions like;
IF “e is less then 0.01” AND “ ˙e is positive” THEN “Set the control signal u to 1.2”
where the propositions “e is less then 0.01” and “ ˙e is positive” constitute the antecedent or the premise part of the IF-THEN statement while the proposition “Set the control signal u to 1.2” is the consequent part. In a statement like “IF p THEN q”, where p and q are are propositions, it is said that p implies q which also has a representation p→ q. The truth value of this implication is given in Table 2.1.
There are many different ways to form and relate a set of propositions. The systematic study of how propositions can be related in ways that have repercussions for their respective truth value is called logic [19]. In [20] (p.29) the application of logic to the basic switching network is given. Also in [18] (p.183) there is an interesting quotation on the same subject which reveals how one can save a small fortune by application of Boolean Algebra, which is one of the tools in logic. We will skip these because our main concern will be the interpretation of the IF-THEN statements using the given sets defined in the antecedent and consequent part. This topic will also be elaborated in Section 3.1.3.
2.2
Models for Switching Systems
Depending on the problem under consideration there are different modelings of switching systems. Let us see some of the systems which are considered as switching systems.
Example 1 Collisions:
Consider the case of the vertical and horizontal motion of a ball of mass m in a room under gravity with constant g. Let x and y denote the horizontal and vertical position of the ball, vx and vy denote their respective velocities. In this case, the dynamics are given
by,
˙x = vx, ˙vx= 0, ˙y = vy, ˙vy =−mg.
Further, upon hitting the boundaries {(x, y)|y = 0 or y = dheight}, vy is instantly set to
−ρvy where ρ ∈ [0, 1]. Likewise, upon hitting {(x, y)|x = 0 or x = dwidth} vx is set to
−ρvx. Here the continuous state x(·) changes impulsively on hitting prescribed regions
of the state space [12].
Example 2 Nonholonomic Integrator :
Consider the nonholonomic integrator with the following dynamics: ˙x = g(x)u(x) g(x) = 1 0 0 1 −x2 x1
Let Ri, i = 1, . . . , 4 be some regions on R3, see e.g. (4.10). In each regionRi the control
signal, u(x), changes as: u = ui(x), when x ∈ Ri. In [21] a set of stabilizing ui(x) is
given.
Example 3 Transmission System :
Consider the dynamics of a transmission system whose simplified model is given below [12],
˙x1 = x2
˙x2 = [−a(x2/v) + u]
where x1 is the ground speed, x2 is the engine RPM, {u ∈ [0, 1]} is the throttle position,
and v ∈ {1, 2, 3, 4} is the gear shift position. The function a is positive for positive argument.
In Example 1 and 3, the change in dynamics can be modelled as the parameter change of the system. The change in the dynamics in Example 2 is due to the change in
the control algorithms depending on the value of the states. There are other examples for different type of switching systems in [12]. As is the case with all modelling issues, a proper model of a switching system should be general enough to encompass a large variety of physical phenomena, yet be structured enough to present a workable model through which interesting problems can be mathematically framed [7]. Although we will restrict our study to the systems where the dynamic changes is due to the change in control algorithm as in Example 2, we will also include the models on which there is any stability result in the literature.
Most of the studies on the stability are performed on linear switching systems. The following dynamics is a common model for most of these studies;
˙x = Aix
i+ = s(x, i) (2.1)
where Ai ∈ Rn×n, x ∈ Rn and s : Rn × I → I and I = {1, . . . , N}. Here s(·) is the
function that determines when to change the value of i which is the index specifying the operation mode of the system. Example 1 can be modelled by Equation (2.1). In (2.1) there is no control signal u. Consider the following dynamics;
˙x = Ax + Bu u = Kix
i+ = s(x, i)
where A ∈ Rn×n, B ∈ Rn×m, x ∈ Rn, and u ∈ Rm. Comparing these equations with
(2.1) we see that Ai = A + BKi and u = ui(x) which is determined by the function s(·)
and K0
is. In a general form the above dynamics can be stated as;
˙x = f (x) + g(x)u u = ui(x)
i+ = s(x, i)
(2.2)
where f (·) : Rn −→ Rn, x ∈ Rn, g(·) ∈ Rn×m, and u ∈ Rm. This model can be used
to express the system given in Example 2. Suppose that the function s(·) in (2.2) is switching to the ith controller when the state x is in a region specified by R
control signal can be expressed with the following equations; u(x) = Pmi=1ui(x)Si(x) Si(x) = 1, x ∈ Ri 0, x /∈ Ri (2.3)
where Si(x) is the characteristic function for the set of points in regionRias in Definition
3. The following dynamics can be used to express the system in Example 3. ˙x = f (x, p)
p+ = s(x, p) (2.4)
where p is the parameter vector. In Example 3 this is equivalent to v, the variable for gear shift position.
2.3
Stability Analysis
Consider the system given by the dynamics (2.1). The stability of a switching system does not only depend on the vector fields Aix but also depends on the order of the
sequence. Since the case where there is no switching is also a valid switching sequence, every matrix Ai must be stable on their own. Note that, there exists stable matrices
A1 and A2 such that under a certain switching strategy the overall system turns out
to be unstable. The converse is also possible, that is there exists unstable matrices A1
and A2 such that under a certain switching strategy the overall system turns out to be
stable [13]. Hence it is necessary to develop certain tools to study the stability of the switching systems when the switching sequence is arbitrary, and to incorporate these tools into the controller design so that the closed-loop system becomes stable. Let us consider the Lyapunov Stability Theorem for a Linear Time Invariant (LTI) system.
Theorem 1 Consider the Lyapunov equation given below; ATP + P A =
−Q (2.5)
A necessary and sufficient condition for a LTI system ˙x = Ax to be strictly stable is that, for any symmetric positive definite matrix Q, the unique matrix P solution of the Lyapunov Equation (2.5) be symmetric positive definite.
Proof 1 See [22]
The following Lemma gives a sufficient condition for a given switching system as in (2.1) be stable for arbitrary switching.
Lemma 1 The switching system (2.1) is exponentially stable if there exists a positive definite matrix P such that V (x) = xTP x is a common quadratic Lyapunov function for
all Ai, i.e.,
AT
i P + P Ai =−Qi (2.6)
for some positive definite matrices Qi [3].
So if one can find a common positive definite matrix for a given set of matrices {A1, . . . , AN} then the stability of the switching system can be proved. Finding a
common positive definite matrix for the Lyapunov function is a hot topic of research see e.g. [3], [23], and [24]. The associated controller design problem is still an open problem and currently under investigation by many researchers. The following subsections define some constraints on the set {A1, . . . , AN} to prove the stability of the switching systems
using Lemma 1.
2.3.1
Commuting Stable Matrices
In [3] the stability of a switching system defined as in (2.1) is considered. The set of matrices {A1, . . . , AN} defining the switching system is assumed to commute pairwise.
The related stability result is given below,
Theorem 2 Consider the switching system in (2.1) with A = {A1, . . . , AN} where the
matrices Ai are asymptotically stable and commute pairwise. Then
i) The system is exponentially stable for any arbitrary switching sequence between the elements of A.
ii) Given a symmetric positive definite matrix P0, let P1, . . . , PN be the unique
symmetric positive definite solution to the Lyapunov equations AT
Then the function V (x) = xTP
Nx is a common Lyapunov function for each of the
individual systems ˙x = Aix i = 1, . . . , N , and hence a Lyapunov function for the
switching system (2.1).
iii) For a given choice of the matrix P0, the matrices {A1, . . . , AN} can be chosen in
any order in (2.7) to yield the same solution PN
vi) The matrix PN can also be expressed in integral form as;
PN = Z ∞ 0 eAT NtN · · · Z ∞ 0 eAT 2t2 Z ∞ 0 eAT 1t1P 0eA1t1dt1 eA2t2dt 2 · · · eANtNdt N (2.8)
where, as in iii) above, the order in which the matrices {A1, . . . , AN} appear can
be replaced by any permutation.
Proof 2 See [3] 2.
2.3.2
Multiple Lyapunov Functions
Consider the system given in (2.1) for N = 2. Let V1(x) = xTP1x and V2(x) = xTP2x
be the two Lyapunov-like functions for the systems defined by ˙x = A1x and ˙x = A2x,
respectively, that is these functions decrease along the trajectories in a certain region. Let us assume that the at time t = 0 the active dynamics was A1. Let the system
switch to the dynamics A2 after τA1,1 seconds. If the system then waits for a duration
of τA2,1 seconds to switch back to A1 we would have a sequence of time durations
like τ = {τA1,1, τA2,1, τA1,2, τA2,2, . . .}. Let tj = tj−1 + τA1,j + τA2,j where t0 = 0. If
one can orchestrate the switching times so that V1(tj−1 + τA1,j+ τA2,j) < V1(tj−1) and
V2(tj+ τA1,j+1) < V2(tj− τA2,j) for all j then the system is asymptotically stable because
of the decreasing tendency. This idea can be generalized to N > 2. There are many research on this subject among which [13] uses multiple Lyapunov functions as a tool for analyzing Lyapunov stability. In [25] the multiple Lyapunov function approach is utilized to stabilize a linear system using finite-state hybrid output feedback and a stabilizing switching sequence for a switched linear system with unstable individual matrices is obtained. This method is utilized in [26] and [27] to determine a switching strategy which is called dwell time switching. This approach will be considered in Section 7.1.1.
2.3.3
Other Stability Results
In this section we will include some other stability results on switching systems. What follows is some theorems stating the stability conditions for some specific structure for the system dynamics.
Triangular Matrices
Theorem 3 If there exists a nonsingular matrix T ∈ Rn×n such that ˜A
i = T AiT−1,
i = 1, . . . , N , are upper-triangular then there exists a positive definite matrix P such that (2.6) holds.
Proof 3 See [23].
Commonly symmetrizable Matrices
Theorem 4 Assume the stable matrices A1, . . . , AN ∈ Rn×n are commonly
symmetriz-able, i.e., there exists a common similarity transformation under which every Ai is
similar to a symmetric matrix. Then the system (2.1) is exponentially stable for arbitrary switching.
Proof 4 See [7].
Theorem 5 Assume the stable matrices A1, . . . , AN ∈ Rn×n are nearly commonly
symmetrizable, i.e., there exists a exist a nonsingular matrix T such that
T AiT−1 = Si+ Ei, Si = SiT, kEik < , > 0. (2.9)
If is sufficiently small, then the switching system, (2.1), is exponentially stable.
N = 2 Case
Theorem 6 Assume that N = 2 and A1, A2 ∈ Rn×n. Then A1, A2 have a
common quadratic Lyapunov function if and only if the matrices αA1+ (1− α)A2 and
αA1+ (1− α)A−12 are stable for all α∈ [0, 1].
Chapter 3
Fuzzy Systems
In our daily life there are many phenomena which can be classified as being certain and uncertain. The class of uncertain phenomena can be classified once more as random and fuzzy. Therefore, we have three categories of phenomena and their associated mathematical models [29]:
1. Deterministic Mathematical Models: This is a class of models where the relationships between objects are fixed or known with certainty. According to our current theory the relation between the distance for a falling object with mass m on Earth and the elapsed time is given by the formula h = 0.5gt2. This is an
example for a deterministic mathematical model.
2. Random (Stochastic) Mathematical Models: This is a class of models where the relationships between objects are uncertain in nature. In other words it is hard to formulate the relationship between objects using an equation. Instead there is the observations given as a distribution. The relation between the outcome of rolling of dies might be an example for an event that can be modelled by random mathematical model.
3. Fuzzy Mathematical Models: This is a class of models where objects and relationships between objects are subjective and expressed with words that are vague to define. Think of someone who describes a person being tall. S/He might not be able to give a precise measure in meters where the boundary for a person
to be tall starts. Even s/he gives a measure it will be a subjective decision and might well be different for someone else.
A fuzzy system is a real time expert system implementing a part of a human operator’s or process engineer’s expertise which does not lend itself to being easily expressed in a difference/differential equation but rather in situation/action rules. However, fuzzy control differs from main-stream expert system technology in several aspects. One main feature of fuzzy control systems is that there are symbolic IF-THEN rules and qualitative, fuzzy variables and values such as:
if “pressure is high” and “slightly increasing” then “energy supply is negative medium”. Most of the researchers in the area of fuzzy control have a strong control engineering and systems theory background. From their perspective, fuzzy control can be seen as a heuristic and modular way of defining nonlinear, table based systems. Reconsider the rule above : it is nothing but an informal “nonlinear Proportional Derivative(PD)-element”.
A representation theorem, mainly due to Kosko [9], states that any continuous nonlinear function can be approximated as closely as needed with a finite set of fuzzy variables, values, and rules. Later X.-J. Zeng and M. G. Singh published their work on the approximation problem of SISO and MIMO fuzzy systems (see [10], [11]). These results describe the representational power of fuzzy control in principle, but it does not answer the questions of how many rules are needed and how they can be found.
It is generally agreed that an important point in the evolution of the modern concept of uncertainty was the publication of a seminal paper by Lotfi A. Zadeh [8]. In this paper, Zadeh introduced a theory whose objects-fuzzy sets-are sets with boundaries that are not precise. The membership in a fuzzy set is not a matter of affirmation or denial, but rather a matter of degree.
The literature in fuzzy control has been growing rapidly in recent years, making it difficult to present a comprehensive survey of the wide variety of applications that have been made. Historically, the important milestones in the development of fuzzy control may be summarized as shown in Table 3.1, [30], [31]. The first successful
1972 Zadeh A rationale for fuzzy control [32] 1973 Zadeh Linguistic approach [33]
1974 Mamdani & Assilian Steam engine control [34]
1976 Rutherford et al. Analysis of control algorithms [35]
1977 Ostergaard Heat exchanger and cement kiln control [36] 1977 Willaeys et al. Optimal fuzzy control [37]
1979 Komolov et al. Finite automation [38]
1980 Tong et al. Wastewater treatment process [39] 1983 Takagi and Sugeno Derivation of fuzzy control rules [40] 1984 Sugeno and Murakami Parking control of a model car [41] 1985 Togai and Watanabe Fuzzy chip [42]
1986 Yamakawa Fuzzy controller hardware system [43] 1988 Dubois and Prade Approximate reasoning [44]
Table 3.1: Some important studies in fuzzy control.
industrial application of the Fuzzy Logic Controller (FLC) was a cement kiln control system developed by the Danish cement plant manufacturer F. L. Smidth in 1979.
FLCs can be used in various ways. One of them is to use the FLC directly as the controller for the system under consideration. The other choice is to use a standard controller like PID or lead-lag and tune the parameters of this controller with a Fuzzy Logic System (FLS). A few of the studies on parameter tunings are given in [31], [45], [46] and [47]. In [46] the PID parameters are tuned according to some rules derived by considering a typical system output. In [47] some changes to the model of fuzzy PI controller is introduced to reduce the overshoot even more. In [31] a heuristic approach is presented to design the fuzzy controllers which tune the parameters of a PID and a Lead-Lag controller.
In the following sections we will first give some necessary definitions and background to analyze fuzzy systems. After that a short overview will be presented on how fuzzy set theory is used in implications to determine the relations between some variables of interest. Next the class of fuzzy systems and their mathematical representation will follow.
3.1
Fuzzy Set Theory
Fuzzy set theory can be seen as the extension of classical set theory in many aspects. In classical set theory the characteristic function which determine whether an object belongs to a set or not is determined depending on the value of that function being either 0 or 1. Let us repeat the definition of characteristic function given in the previous chapter. Let X be the universe of discourse, and let A ⊂ X. Then the characteristic function, µA : X → {0, 1}, of the set A is defined as:
µA(x) = 1, x∈A 0, x /∈A
Fuzzy sets on the other hand fills the gap for sets with boundaries that can not be defined precisely. To motivate fuzzy reasoning, let us try to define the sets described by the phrases like “tall man”, or “hot weather”. An appropriate characteristic function for such sets described by such vague expression can be obtained by expanding its range to the closed interval [0, 1]. The characteristic function of a fuzzy set is called the membership function (MF). In Figure 3.1 the 3 mostly encountered membership function profiles are given. These membership functions are chosen to have different mean values for their support which will be defined later. This choice is for better visualization.
−1.5 −1 −0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 Triangular MF Gausian MF Trapezoidal MF
Figure 3.1: The general structure of a fuzzy systems.
In Figure 3.1, the horizontal axis represents the quantity of the concept being defined. For example the temperature, which can be measured, might be the horizontal axis
in defining the membership function of the set ”hot weather”. We will represent the membership functions of the fuzzy set S by µS(·).
If the universe of discourse of the membership function is continuous as in the “temperature”, then the function can be represented as a continuous or piece-wise continuous functions as in Figure 3.1. In most of the cases the control algorithms will be implemented by one or the other digital computing device which have a finite number of bit resolution. Hence even if we have measurements of temperature, its values after passing from an analog to digital converter will be quantized to a discrete universe of discourse. If the domain is discrete then we have tuples to represent fuzzy sets. Interested readers can resort to [30] or any related book such as [17], [48] for definitions fuzzy sets/relations whose universe of discourse is discrete and for more information on fuzzy systems. Here we will give the definition and examples of membership functions for continuous universe of discourse.
Definition 6 Support :
The support of a function, µS(·) : X −→ [0, 1] is the set {x|µS(x) > 0} 2.
Definition 7 Convex Function:
Let x1, x2 ∈ X = R be any two elements within the support of the function,
µS(·) : X −→ [0, 1]. If µS(αx1+ (1− α)x2)≥ αµS(x1) + (1− α)µS(x2) is satisfied
∀ x1, x2, ∀ α ∈ [0, 1] then the function µS(·) is convex. This definition can be extended
to X = Rn. 2
Definition 8 Membership Function :
The membership function µS(·) of a fuzzy set S is a function µS(·) : X −→ [0, 1]. The form of a membership function can be arbitrary but as a general practice, it is required to have a convex membership function with finite support to ease certain manipulations 2.
In Figure 3.1 some of the membership function profiles are given. These membership functions can be expressed as in (3.1) where m is the point the membership function is equal to 1 and w is a measure of the spread of the support. T g stands for triangular,
Ga for Gaussian, and T z for trapezoidal membership function. While a and d are the borders of the support, b and c determine the interval where the trapezoidal membership function is 1. µT g(x) = 1− |x−m|w , |x − m| < w 0, |x − m| ≥ w µGa(x) = e −|x−m|2 2w2 µT z(x) = 0, x < a x−a b−a, a < x < b 1, b < x < c x−d c−d, c < x < d 0, d < x (3.1)
A fuzzy set S might be defined as those elements in X where the associated membership function µS(x) > 0 i.e. S = {x|µS(x) > 0, x∈ X } . In fact those elements having a membership value of zero is not in the set. By defining the fuzzy set as S = {x|µS(x) > 0, x∈ X } we make no difference with the definition of a classical set
made by the use of characteristic function and have ignored the graded membership value of the element to the set. Therefore it is best to define a fuzzy set as a collection of tuples.
Definition 9 Fuzzy Set [17]:
Let X be the universe of discourse and S ⊂ X be a set which has membership function is µS(·) : X −→ [0, 1]. Then the fuzzy set S is defined as a collection of tuples given as:
S = {(x, µS(x))|x ∈ X }
There are many different approaches to determine the membership function of a fuzzy set. These approaches depend mainly on the problem at hand. Depending on the problem one can use the methods which are based on frequency of the occurrence of the object as in [29] or use some clustering algorithms like fuzzy-c means as in [31]. Some other approaches can be found in [49]. All of these different approaches lead us to the
following definitions which are used to classify fuzzy sets so that some of the operations are simplified.
Let us give some more definitions on membership function which will be used in the following sections. Let In denote the set In ={1, . . . , n}.
Definition 10 Normal Membership Function:
Let S be a fuzzy set defined by the membership function, µS(·) : X −→ [0, 1], i ∈ In.
Then the membership function, µS(·), is said to be normal if sup µS(·) = 1.2
Definition 11 Completeness :
Let S1, . . . ,Sn be fuzzy sets defined by the membership functions, µSi(·) : X −→ [0, 1],
i∈ In. Then these fuzzy sets are said to be complete on X if for any x ∈ X , there exists
Si such that µSi(x) > 0.2
Definition 12 Consistency :
Let S1, . . . ,Sn be fuzzy sets defined by the membership functions, µSi(·) : X −→ [0, 1],
i∈ In. The fuzzy sets are said to be consistent if µSi(x0) = 1 for some x0 ∈ X , then for
all j 6= i µSj(x0) = 0.2
Another problem in fuzzy system design is the optimization of the functional form. In these approaches the membership functions are assumed to have a predefined structure as in Figure 3.1. After the parametrization of these membership functions, the parameters are optimized to minimize a predefined cost function which might aim to reduce the number of partitions of the universe of discourse or aim to increase the performance of the system.
3.1.1
Operations on Fuzzy Sets
The expression, “A AND B” in propositional logic is true if and only if both expressions A and B are true. In fuzzy set theory their interpretation is not so simple, because graded characteristic functions (membership function ) are used. Zadeh proposed [8]
the following definitions for the union, intersection and complement operations on fuzzy sets.
Definition 13 Let X be the universe of discourse, A, B ⊂ X be two fuzzy sets and let µA(·) and µB(·) be their membership functions, respectively. Then the membership
function of the sets A ∪ B, A ∩ B, Ac may be given as follows :
∀x ∈ X : µA∩B(x) = min(µA(x), µB(x)),
∀x ∈ X : µA∪B(x) = max(µA(x), µB(x)),
∀x ∈ X : µAc(x) = 1− µA(x). 2
If we update the list of set operations for classical sets given in the Chapter 2, with the definition of Zadeh we have the following list.
Fuzzy SetA, A = {(x, µA(x))|x ∈ X } Fuzzy SetB, B = {(x, µB(x))|x ∈ X }
Fuzzy SetU, U = {(x, µU(x))|x ∈ X } Complement ofA, A0 = {(x, 1 − µ
A(x))|x ∈ X }
Inersection of A and B, ATB = {(x, min{µA(x), µB(x)})|x ∈ X } Union of A and B, ASB = {(x, max{µA(x), µB(x)})|x ∈ X }
Difference of A from B, A − B = {(x, max{µA(x), 1− µB(x)})|x ∈ X } Symetric difference of A and B, A + B = (A − B)S(B − A)
Power set ofA, P (A) = {(x, µU(x))|µA(x)) ≥ µU(x), x∈ X } Cartesian product ofA and B, A × B = {(x1, x2, min{µA(x1), µB(x2)})|x ∈ X }
Power n ofA, An = A × · · · × A
| {z }
ntimes
Note that in the definition of the cartesian product of the set A and B, a fuzzy set defined by A × B is not tuple but triple where the membership function defining the product has a domain of product space. This will be clear in Section 3.1.2 in which we give the definition of a relation. If the values of µA(x) and µB(x) are restricted to the set{0, 1} then the results reduce to the classical set operations. Therefore this is a very simple extension of the classical set operations. There are other extensions, for example,
∀x ∈ X : µA∩B(x) = µA(x)· µB(x),
There is a class of operators called t-norm for intersection operations and another called s-norm for union operations. Readers can resort to [17], [30], and [48] for a more comprehensive literature survey on fuzzy sets and operations defined on them.
3.1.2
Fuzzy Relations
In Chapter 2 we have described the classical relation as being a classical set in a product space. This applies also for fuzzy relations. As we can define fuzzy sets with membership functions, so we can do with fuzzy relations.
Definition 14 Fuzzy Relation :
Let U × V be the universe of discourse and µR : U × V −→ [0, 1] be the membership
function of a relation R. (As in the case of fuzzy sets the membership function of a relation can have any arbitrary form.) Then the fuzzy relationR is defined as a collection of triple as:
R = {(u, v, µR(u, v))|u ∈ U and v ∈ V}
This definition can be extended to relations whose membership function is defined as µR:U1 × · · · × Un−→ [0, 1]. 2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 v u µR (u,v)
